Transformation texture and Orientation relationship

Transformation texture and
Orientation relationship
Vahid Tari, Ben Anglin, S. Mandal, and A.D. Rollett
27-750
Texture, Microstructure & Anisotropy
Lastrevised28thApr.2016
2
Objec8ve
•  Understand orientation relationships
•  Phase transformation and variant selection
in steel alloys
•  Phase transformation and variant selection
in Titanium alloys
•  Construct misorientation matrices from given
orientation relationship
•  Forward and backward texture prediction
3
Orienta8onRela8onships
•  Orientation Relationship (OR): Relation between
specific planes and directions of two crystals on
either side of boundary.
•  During most phase transformations, some
favored orientation relationship exists between
the parent and the product phases which allows
the best fit at the interface between the two
crystals.
•  Why important? Phase transformations;
morphology of precipitates; nucleation
mechanisms; interfaces; high temperature
orientation determination; thin film orientation
prediction.
Why transformation texture and OR
are important?
• 
Ray and Jonas, International Materials Reviews 1990 Vol. 35 NO.1
• 
Ray and Jonas, International Materials Reviews 1990 Vol. 35 NO.1
6
Orienta8onrela8onship(OR)
Notation:
(hkl)α//(h’k’l’)β
[uvw]α//[u’v’w’]β
The (hkl) plane of the α crystal lies parallel to the (h’k’l’)
plane of the β crystal
Similarly for the [uvw] and [u’v’w’] directions of the two
crystals
[uvw] and [u’v’w’] must lie in the (hkl) and (h’k’l’) planes,
respectively
The orientation relationship holds regardless of the
coherency of the boundary
OR
Variants
Variants of a given OR are specific alignment of planes and directions. These exist
because of crystal symmetry
In the K-S OR, there are 4 {111}γ planes, each plane parallel to a {110}α plane. A
{111}γ plane contains 3 〈110〉γ directions and each 〈110〉γ direction is parallel to 2
〈111〉α directions [1]. Hence 24 K-S variants.
[1
1 1]γ || [0 1 1]α
[1
[1
0 1 γ || 1 1 1 α
2
] [
1] || [2
γ
[1]Verbeken,Barbe,Raabe,ISIJInt,2009(49)10,1601-1609
1
]
1]
α
Butron-Guillen,DaCostaViana,Jonas,MetTransA,1997(28A)1755-1768
7
Orienta8onrela8onshipsinSteel
Stereographic Projection of ORs
010
100
{100}
Variants Selection
•  Variants selection: Few of the theoretically
predicted variants dominant. Some variants may
be preferred over others depending on the
transformation mechanism.
•  Knowledge about variants selection important to
understand microstructure evolution.
Fig: Evolution of microstructure during diffusional phase transformation (a) without
variant selection (each nucleus with a different variant becomes a different grain,
resulting in a fine structure). (b) with variant selection (neighboring nuclei having the
same variants coalesce to form larger grains). (Furuhara and Maki, 2001)
10
Variantselec8onduringphase
transforma8oninsteel
• 
Youliang He, John J Jonas, Stéphane Godet, Metallurgical and Materials Transactions A, Vol. 37A, 2006,2641
OR (Burgers) in Titanium Alloys
Phasetransforma8on
BCC
HCP
•  Predominant OR
observed for bodycentered cubic to
hexagonal
transformation in Ti
alloys
•  Burgers OR
(0 0 0 1)HCP ||{0 1 1}BCC
[1 1 -2 0]HCP||⟨1 1 1⟩BCC
Fig: Geometrical representation of the
Burgers OR with dashed lines showing the
BCC crystal and continuous lines showing
the HCP crystal (Menon et al.,1986)
12
13
Burgers OR Variants
•  Variants: Each
crystallographic orientation
relationship predicts different
numbers of product
orientations originating from a
single crystallographic
orientation of the parent
phase.
•  Number of variants is
determined by the OR that the
product phase have with the
parent phase.
•  # variants for Burger’s OR=12
14
Stereographic Projection of Burgers OR
Plane alignment
Stereographic projection for one
of the variants of the Burgers OR
with a parent BCC orientation of
{0,0,0}. Constructed using inhouse scripts OR stereogram and
DrawPF. Available from AD Rollett.
BCC (high T)
Direction alignment
Hexagonal
(low T)
111
0001
100
11-20
110
10-10
15
An example
Fig: IPF map of
an EBSD
dataset.
(Provided by
IISc)
Fig: Pole figures
generated using
TSL software for
this data.
16
Variants Selection in Ti
150
140
130
120
[-1 1 -1]
110
Total area
100
[-1 1 1]
90
80
[ 1 1 -1]
70
60
MEAN
[ 1 1 1]
50
40
ß Directions
30
20
10
0
1
2
3
4
5
6
7
8
9
10
11
12
Variant
(110)
(1-10)
(011)
(01-1)
(101)
(-101)
ß Planes
Fig: Relative frequency of variants for the EBSD dataset (Provided by IISC).
17
Creating Misorientation Matrix for OR
[u
v w]α || [u ' v' w']β
(h
k l )α || (h' k ' l ')β
Assume:(hkl)//(h’k’l’)//ND
and[uvw]//[u’v’w’]//RD
⎛ b1 t1 n1 ⎞
⎜
⎟
gα = ⎜ b2 t 2 n2 ⎟
1.)Constructorienta8onmatrixforphaseα.
⎜b t n ⎟
3⎠
⎝ 3 3
⎛ b'1 t '1 n'1 ⎞
2.)Constructorienta8onmatrixforphaseβ.
⎜
⎟
g β = ⎜ b'2 t ' 2 n'2 ⎟
⎜ b' t ' n' ⎟
3
3⎠
⎝ 3
3.)Computemisorienta8onmatrixfromtheαphasetotheβphase.
Notetheimportanceofthesenseofthetransforma8on;forgrain
boundariesweinvokeswitchingsymmetrybutyoumustnotapplythis
tophasetransforma8ons!
Δg = g β gα−1
18
Creating Misorientation Matrix for OR
ConsidertheK-Sorienta8onrela8onship:
(111)fcc||(-110)bcc
[1-10]fcc||[111]bcc
1) Construct orientation matrix for phase fcc.
2) Construct orientation matrix for phase bcc.
3) Compute misorientation matrix from one phase to the other; here we pass
from fcc (high temperature) to bcc (low temperature).
Minimum misorientation angle and axis
associated with ORs
T =
g = Obcc gbcc (Of cc gf cc )
T =
1
g = Obcc gbcc gf cc
Of cc
1
Minimum misorientation angle and axis
associated with ORs
ORs in Rodrigues space
Orienta8onsandRota8ons,AdamMorawiec,2003
S.-B.Leeetal./ActaMaterialia60(2012)1747–1761
KS Orientation relationship
in Rodrigues Space
GT Orientation relationship
in Rodrigues Space
GT’ Orientation relationship
in Rodrigues Space
NW Orientation relationship
in Rodrigues Space
Pitsch Orientation relationship
in Rodrigues Space
Notethatthe~45°anglemeansthatthe
pointsliealmostonthesurfacesofthe
Rodriguesspaceforcubic(mis)orienta8ons
Forward Texture Prediction
Phasetransforma8on
Parent
(FCC)
Daughter
(BCC)
KS
Tari et al. (2013) J Appl. Cryst., 46 210-215.
NW
Backward Texture Calculation
Daughter
(BCC)
Parent
(FCC)
v1
v6
v2
p1
v5
gdj= BCC variants orientation
Qi= symmetry operator
T= OR matrix
gpi= FCC variants orientation
p2
v4
v3
p3
Tari et al. (2013) J Appl. Cryst., 46 210-215.
Backward Texture prediction
α1α2α3α4αn
BCC
variants
FCC possible variants
Vγ1_1 Vγ2_1
Vγ1_i
Vγ1_n
Vγ3_1 Vγ4_1
Vγ3_i Vγ4_i
Vγ2_n Vγ3_n Vγ4_n
Vγ2_i
Vγn_1
Vγn_i
Vγn_n
Tari et al. (2013) J Appl. Cryst., 46 210-215.
Backward Texture prediction
FCC possible variants
BCC
variants
α1α2α3α4αn
Vγ1_1
Vγ1_i
Vγ1_n
Vγ2_1
Vγ2_k
Vγ2_i
Vγ2_n
Vγ3_1
Vγ3_i
Vγ3_n
Vγ4_1
Vγ4_i
Vγ2_j
Vγ4_n
SMMA_γ1_1
SMMA_γ2_i
SMMA_γ2_n
Tari et al. (2013) J Appl. Cryst., 46 210-215.
Backward Texture prediction
Austenite
KS
Bainite
NW
Backward Texture prediction
Tari et al. (2013) J Appl. Cryst., 46 210-215.
Backward Texture prediction
0.04C–1.52Mn–0.2Si–0.22Mo–0.08Ti–0.033Al (in wt.%)
Beladi et al. (2014) Acta Materialia 63 86–98
34
Orientation Relationships in Pearlite Microstructures
Pearlite is a lamellar structure comprised of BCC α-ferrite and orthorhombic1 cementite (Fe3C). A
schematic of the pearlitic lamellar structure is shown below, with many different length scales represented:
prior austenite grain size, pearlite colony size, interlamellar spacing, and cementite thickness. If a high
enough microstructural resolution is used, all length scales should be visible in the vpFFT simulations. This
may make a case for the use of multiple SVEs rather than RVEs for this microstructure. (Something to be
decided later.) Additionally, because of the sharp and (usually) straight ferrite and cementite interface, there
likely exists an orientation relationship (OR) between the two phases. In fact, three ORs have been reported
in the literature2. Two of these ORs occur with a higher frequency than the third, and these two occur with
the same frequency. The three ORs are Bagaryatsky3, Isaichev4, and Pitsch-Petch5,6 and are described on the
lower right. The Isaichev OR is the least frequently observed OR.
Bagaryatsky:
[100]c//[1͞10]f
[010]c//[111]f
(001)c//(͞1͞12)f
Isaichev:
[010]c//
[111]f
(101)c//(11͞2)f
Pitsch-Petch:
[100]c 2.6 deg from [͞31͞1]f
[010]c 2.6 deg from [131]f
(001)c//[21͞5]f
1I.G.
Wood et al., J Applied Crystallography, 37 (2004) 82-90
Mangan, G.J. Shiflet, Met Trans A, 30A (1999) 2767-81
3Y.A. Bagaryatsky, Dokl Akad Nauk SSSR, 73 (1950)1161-64
4I.V. Isaichev, Z Tekhn Fiz, 17 (1947) 835-38
5W. Pitsch, Acta Cryst, 10 (1962) 79-80
6N.J. Petch, Acta Cryst, 6 (1953) 96
2M.A.
A.M. Elwazri et al.,
Mat Sci and Eng A,
404 (2005) 91-98
35
Further Literature Review of Cementite ORs
Why do different ORs exist for ferrite and cementite in pearlite?
In a large review paper on using electron backscatter diffraction for the study of phase transformations in
20021, a section pertaining to pearlite was presented. It cited a 1999 paper2 which found that the PitschPetch orientation relationship occurs when the pearlite colony nucleates first with cementite. The
Bagaryatsky orientation relationship was observed when the pearlite colony nucleated with ferrite. This was
observed in both hypereutectoid or hypoeutectoid alloys. All results of ORs were confirmed using EBSD.
A 2009 review paper3 on predicting orientation relationships using an edge-to-edge matching method to
minimize the misfit at an interface stated that using a “selected area electron diffraction” is insufficient
resolution to differentiate between the Isaichev and Bagaryatsky ORs. Higher resolution measurements
using convergent beam Kikuchi line diffraction patterns never observed the Bagaryatsky OR. They go as far
as to say the Pitsch-Petch OR also does not exist, but it is really four distinct ORs which vary less than 6o
from Pitsch-Petch. Additionally, the Bagaryatsky and Isaichev OR vary by about 3.5o.
However, in a 2008 PhD thesis4 the Bagaryatsky OR was observed using EBSD, which has a angular
resolution of about 1-2 degrees.
With the small difference between ORs and EBSD confirmation of the Bagaryatsky, I still believe that the
Bagaryatsky OR (along with the Pitsch-Petch OR) is still valid and occur about 50/50.
1Gourges-Lorenzen, A.F.,
Int. Materials Reviews, 52 (2002) no. 2
Shiflet, Met Trans A, 30A (1999) 2767-2781
3Zhang, M.X., Kelly, P.M., Progress in Materials Science, 54 (2009) 1101-1170
4Nikolussi, M., PhD Thesis, 2008
2Mangan,
Pearlite: Hypothesis and Experiments
36
Hypothesis: Pearlite colonies containing the Bagaryatsky OR will deform more than those containing the Pitsch-Petch OR.
Concise background relating to hypothesis:
Orientation Relationships
Bagaryatsky:
[100]c//[1͞10]f
[010]c//[111]f
Habit Planes → (001)c//(͞1͞12)f
Pitsch-Petch:
[100]c 2.6 deg from
[͞31͞1]f
[010]c 2.6 deg from [131]f
Habit Planes → (001)c//[21͞5]f
Bagaryatsky OR
With the slip plane in ferrite
aligned to the habit plane, long
distance between obstacles on
this slip plane
Slip Systems of Pearlite
Ferrite:
<111>{110}
<111>{112}
Cementite:
<100>{001}
<100>{011}
<111>{110}
Isotropic Elevated Yield Stress
σ y = σ 0 + k y S0−1
Pitsch-Petch OR
With the slip plane in ferrite
misaligned to the habit plane,
short distance between obstacles
on the {112} slip planes. In fact,
{1-21} is perpendicular to {125}
Pearlite: Hypothesis and Experiments
37
Hypothesis: Pearlite colonies containing the Bagaryatsky OR will deform more than those containing the Pitsch-Petch OR.
Concise background relating to hypothesis:
Orientation Relationships
Bagaryatsky:
[100]c//[1͞10]f
[010]c//[111]f
Habit Planes → (001)c//(͞1͞12)f
Pitsch-Petch:
[100]c 2.6 deg from
[͞31͞1]f
[010]c 2.6 deg from [131]f
Habit Planes → (001)c//[21͞5]f
Bagaryatsky OR
With the slip plane in ferrite
aligned to the habit plane, long
distance between obstacles on
this slip plane
Slip Systems of Pearlite
Ferrite:
<111>{110}
<111>{112}
Cementite:
<100>{001}
<100>{011}
<111>{110}
Isotropic Elevated Yield Stress
σ y = σ 0 + k y S0−1
Pitsch-Petch OR
With the slip plane in ferrite
misaligned to the habit plane,
short distance between obstacles
on the {112} slip planes. In fact,
{1-21} is perpendicular to {125}
Pearlite: Varying Fractions of the ORs
Testing of this hypothesis is to be performed using the viscoplastic FFT simulations. I am building seven (7)
different microstructures from a single parent ferrite grain microstructure. These seven microstructures are:
1.) 0_B-100_PP – 100% of all parent ferrite grains spawn cementite lamellae with the Pitsch-Petch OR
2.) 25_B-75_PP – 75% of all parent ferrite grains spawn cementite lamellae with the Pitsch-Petch OR, the other
25% of all parent ferrite grains spawn cementite lamellae with the Bagaryatsky OR
3.) 50_B-50_PP – 50% of all parent ferrite grains spawn cementite lamellae with the Pitsch-Petch OR, the other
50% of all parent ferrite grains spawn cementite lamellae with the Bagaryatsky OR
4.) 75_B-25_PP – 25% of all parent ferrite grains spawn cementite lamellae with the Pitsch-Petch OR, the other
75% of all parent ferrite grains spawn cementite lamellae with the Bagaryatsky OR
5.) 100_B-0_PP – 100% of all parent ferrite grains spawn cementite lamellae with the Bagaryatsky OR
6.) Polycrystalline – this microstructure is the parent ferrite grains with a number of ferrite grains switched phase
switched to cementite (from ferrite) to match the 16% volume fraction of cementite in pearlite
7.) Ferrite only – this microstructure is simply the parent ferrite grains
Recall that there is no true length scale involved in the FFT simulations. So the simulations are not informed as
to the length to the nearest boundary which will impede slip on a given slip system. That is to say, these tests will
simply test if the introduction of different ORs in the microstructure play a role in the deformation behavior of
pearlite. Additionally, comparison of the lamellar microstructure to the polycrystalline sample accentuates the
role of the different ORs compared to a simple incorporation of cementite in to the system.
ORs are assigned in the order in which the parent grains are randomly selected making the assignment of OR to
each grain random. Additionally, the axis along which the OR is place (the (-1-12) in the case of Bagaryatsky) is
also done with a random symmetry operator (belonging to the cubic set).
38
39
while inserted_twins < desired_number_of_twins AND unfilled_grains?!
!
Randomly select grainID!
!
Get grain orientation
!gi!
!
while room for twin within grainID!
!
What is the habit plane normal, vc, in crystal reference frame?!
!
Choose random crystal symmetry operator for crystal class of
!
!parent grain
!Oc(n)!
!
Apply crystal symmetry
!
!
!vc’ = Oc(n) vc!
Find direction in sample space
!vs = g1T vc’!
Construct grain geometry of twin/lamella within parent grain!
!
Define rotation from parent orientation to twin/lamella
!orientation, Δg, by orientation relationship!
!
Assign lamella twin/orientation gt by
!
!
!gt = Δg Oc(n) gi!
!
More twins to insert within grain? !
!
done!
!
All grains filled with twins?!
!
More twins to insert overall? !
!
done!
!!
!
Pearlite: Varying Fractions of the ORs
Grain Structure Images
Ferrite Only
Polycrystalline
Colored by grain number
Polycrystalline – Distribution of Cementite Phase
Colored by phase – red = cementite
40
Pearlite: Varying Fractions of the ORs
Grain Structure Images (Cont’d)
0_B-100_PP
25_B-75_PP
41
50_B-50_PP
Colored by grain number
dark red = cementite grains
75_B-25_PP
100_B-0_PP
Pearlite: Varying Fractions of the ORs
Grain Structures
Structure
Recall that the Fe-C equilibrium phase
diagram predicts the weight fraction of
cementite be about 11%. The table on
the right displays the measured volume
fraction (for simplicity, assumed to be
equivalent to weight fraction) of the
cementite in each microstructure.
The number of twins inserted in to the
51 parent ferrite grains are also listed.
0_B-100_PP
25_B-75_PP
Vol.Frac.Cemen1te
42
No.ofTwins
Inserted
0_B-100_PP
0.124
381
25_B-75_PP
0.126
378
50_B-50_PP
0.123
373
75_B-25_PP
0.125
377
100_B-0_PP
0.124
380
Polycrystalline
0.125
0
FerriteOnly
0.000
0
Fe-CEquilibriumPhaseDiagram
0.1105
——
50_B-50_PP
75_B-25_PP
100_B-0_PP
Ferrite Only
Polycrystalline
Pearlite: Varying Fractions of the ORs
Stress Strain Curves
These are the overall stress-strain curves
for the tested microstructures. The
variation in stress response in the
pearlitic microstructures no longer
increases with increasing Pitsch-Petch
orientation relationship.
Notice the lowest stresses are found in
microstructures without cementite.
Introduction of cementite just as a
second phase (polycrystalline) only
slightly increases by 200 MPa.
The restriction of cementite to a lamellar
structure additionally increases stress
values. On average, this is an increase of
350 MPa for the 100_B-0_PP
microstructure.
The average difference between the
100_B-0_PP and 25_B-75_PP curves is
80 MPa.
43
Pearlite: Varying Fractions of the ORs
Stress Strain Curves
To test the likelihood of the orientation
playing a dominating role on the
deformation behavior, I took the original
50_B-50_PP grain structure and
assigned orientations to all grains at
random. I did this 10 times.
The results to the right show the range
of stress response possible for various
orientations given this grain structure.
The average difference between the
highest and lowest stress-strain curves is
~40 MPa.
Therefore, it seems reasonable that some
differences in response for the
microstructures is due to the structure
itself. (i.e. the spatial orientations of the
lamellae, which is somewhat controlled
by the parent ferrite orientation)
44
Texture Development
0% Strain – Cementite
45
TD
0_B-100_PP
25_B-75_PP
50_B-50_PP
RD
75_B-25_PP
100_B-0_PP
Texture Development
100% Strain – Cementite
46
TD
0_B-100_PP
25_B-75_PP
50_B-50_PP
RD
75_B-25_PP
100_B-0_PP
TD
0_B-100_PP
25_B-75_PP
50_B-50_PP
47
Texture Development
100% Strain – Ferrite
RD
75_B-25_PP
100_B-0_PP
48
Partitioning by OR
Non-random ferrite orientations
0_B-100_PP
25_B-75_PP
50_B-50_PP
No matter the fraction of the OR
present, grains (ferrite and cementite)
associated with the Pitsch-Petch OR
contain a higher stress value on average
than those associated with the
Bagaryatsky. This agrees with the
hypothesis.
For 0_B-100_PP and 100_B-0_PP the
partitioned values match the overall
curve (obviously).
75_B-25_PP
100_B-0_PP
Partitioning by Phase
Non-random ferrite orientations
Ferrite
49
Cementite
It is interesting to see that regardless of the OR fraction, the stress values in ferrite do not vary much. However, the
cementite stress values seem quite dependent on the OR. While the overall response of the pearlite agrees with the
hypothesis, my reasoning seems wrong. I reasoned that in the Bagaryatsky OR the ferrite slip-plane is highly aligned with
the interface plane allowing for dislocations to move longer distances prior to encounter obstacles. Currently, I do not
have reasoning as to why cementite, instead of ferrite, is sensitive to the OR.
50
Microstructure of Martensite
•  The microstructural characteristics of martensite
are:
- the product (martensite) phase has a well defined
crystallographic relationship with the parent (matrix).
- martensite forms as platelets within grains.
- each platelet is accompanied by a shape change
- the shape change appears to be a simple shear
parallel to a habit plane (the common, coherent
plane between the phases) and a uniaxial expansion
(dilatation) normal to the habit plane. The habit
plane in plain-carbon steels is close to {225}, for
example (see P&E fig. 6.11).
- successive sets of platelets form, each generation
forming between pairs of the previous set.
- the transformation rarely goes to completion.
51
Microstructures
Martensite formation
rarely goes to completion because
of the strain associated
with the product
that leads to back stresses in the
parent phase.#
Shear strain in martensite formation
•  The change in shape that occurs during martensite formation is important to
understanding its morphology.
•  In most cases there is a large shear strain. This shear strain is, however,
opposed by the surrounding material.
•  A typical feature of martensitic transformations is that each colony of
martensite laths/plates consists of a stack in which different variants
alternate. This allows large shears to be accommodated with minimal
macroscopic shear. The reason for this morphology is that the volume of
matrix affected by the sheared material is minimized by this alternating
pattern of laths.
52
Atomic model - the Bain Model
53
•  For the case of fcc Fe transforming to body-centered tetragonal (bct) ferrite
(Fe-C martensite), there is a basic model known as the Bain model.
•  The essential point of the Bain model is that it accounts for the structural
transformation with a minimum of atomic motion.
•  Start with two fcc unit cells: contract by 20% in the z direction, and expand
by 12% along the x and y directions.
Orientation relationships in the Bain model are:
(111)γ <=> (011)α’
[101]γ <=> [111]α’
[110]γ <=> [100]α’
[112]γ <=> [011]α’
NBThefccla7cecanbe
obtainedfromthebcc
la7cebyexpandinginthe
verAcaldirecAonbya
factorof√2.
54
Crystallography, contd.
•  Although the Bain model explains several basic aspects of
martensite formation, additional features must be added for
complete explanations (not discussed in detail here).
•  The missing component of the transformation strain is a
rotation and an additional twinning shear that changes the
character of the strain so as to account for the experimental
observation of an invariant [undistorted] plane. This is
explained in figs. 6.8 and 6.9 and the accompanying text.
•  A rather better explanation can be found in Physical Metallurgy
by P. Haasen, pp 337-343. The best approach to the problem
puts it into the form of an eigenvalue equation, with
transformation matrices to describe each of the 3 component
steps of the transformation.
55
Role of Dislocations
•  Dislocations play an important, albeit hard to define
role in martensitic transformations.
•  Dislocations in the parent phase (austenite) clearly
provide sites for heterogeneous nucleation.
•  Dislocation mechanisms are thought to be important
for propagation/growth of martensite platelets or
laths. Unfortunately, the transformation strain (and
invariant plane) does not correspond to simple lattice
dislocations in the fcc phase. Instead, more complex
models of interfacial dislocations are required.
Cu-Nb laminate composites
•  The next few slides provide information about
Cu-Nb composites made by either physical vapor
deposition (PVD), or by accumulative roll
bonding (ARB). This was the subject of a longterm research program at the Los Alamos
National Laboratory.
•  Interestingly, the PVD composites exhibited the
K-S OR with habit plane being the same as the
coincident planes. The ARB composites,
however, exhibited the same K-S OR but with
variable habit plane.
Application where OR is important –
Accumulated Roll Bonding
Radiation damages bulk
crystalline materials#
57
Nanocomposites with high content of
certain interfaces are radiation tolerant#
After 33keV He+ bombardment to ~7dpa#
• InlayeredCu-Nbcomposites,Kurdjumov-Sachsinterfaceplaysanimportantrole,canactas
“supersink”tostoreradia8oninduceddamages,trappingandrecombiningdefectsatthe
interface
• Pointdefectforma8onenergiesareorderofmagnitudelowerandratesofFrenkelpair
annihila8onsignificantlyhigheratinterfacesthanneighboringcrystallinelayers
– Healdamagebytrappingandrecombiningdefectsbeforeclusteringcantakeplace
A.Misra,M.J.Demkowicz,X.Zhang,R.G.Hoagland,JOM60,62(2007)
HIPD Results – ARB
58
Leeetal.(2012)Actamaterialia601747
HIPD=HeterophaseInterphasePlaneDistribu8on
Cu:FCC
Nb:BCC
770nmas-rolled,EBSD
SamplesscannedatLosAlamos.
59
HICD Results - ARB (KS)
Minimumaxis-anglepair,42.85°<0.9680.1780.178>
001standardstereographicprojec1on
010
44.88(maxMRD)
110
-110
011
-111
111
-101
-100
-1-11
45.68(maxMRD)
001
0-11
101
1-11
1-10
-1-10
ActaMater.59(2011)7744;Demkowicz&
Thilly;7°8ltawayfromideal{112}//{112}
lowersinterfaceenergyfrom820downto
690mJ/m-2
100
0-10
770nmas-rolled,EBSD
Ex Situ Thermal stability Study of ARB
Cu-Nb (18nm)
•  Used drop furnace – each sample held for 1 hr at
temperature in vacuum (10-7 Torr) and then furnace
cooled
•  Compare to conventional K-S ({111}Cu//{110}Nb)
interface, {112} K-S is high energy interface, is it
thermally stable?
•  Does the interface deviate from {112} K-S stable?
•  Is a triple junction stable?
•  Is a very thin layer stable?
{112} K-S interface is
thermally stable
Interfaces that deviate from
{112} K-S thermally unstable
Widmanstätten morphology
•  Widmanstätten’s name is associated with
platy precipitates that possess a definite
crystallographic relationship with their
parent phase.
•  Examples:
- ferrite in austenite (iron-rich meteors!)
- γ’ precipitates in Al-Ag (see fig. 3.42)
- hcp Ti in bcc Ti (two-phase Ti alloys,
slow cooled)
- θ’ precipitates in Al-Cu
•  The latter example is based on the
orientation relationship (001)θ’//{001}Al,
[100]θ’//<100>Al. See fig. 3.41 for a
diagram of the tetragonal structure of θ’
whose a-b plane, i.e. (001), aligns with the
(100) plane of the parent Al.
62
63
The Basics
Orientation relationships describe the specific orientation
between two crystals
This is usually done for phase boundaries, but can describe
grain boundaries as well.
Can you think of a special grain boundary that has a wellknown orientation relationship?
(111)α//(111)β
[011]α//[110]β
The Σ3 CSL boundary has this OR. What is the habit plane if
this is a coherent Σ3? Answer: {111}
64
Some common ORs
Bagaryatsky OR
[1 0 0]cem ||[0 -1 1]fer
[0 0 1]cem||[-1 -1 2]fer
Between cementite and
ferrite phases in Pearlite.
Kurdjumov-Sachs OR
{111}FCC||{0 1 1}BCC
<101>FCC||⟨1 1 1⟩BCC
Between austenite and ferrite
phases in Iron; Cu-Zn.
Burgers OR
(0 0 0 1)HCP ||{0 1 1}BCC
[1 1 -2 0]HCP||⟨1 1 1⟩BCC
Between alpha and beta
phases in Ti, Zr.
Potter OR
(0 1 -1 1)HCP ||{-1 0 1}BCC
[2 -1 -1 0]HCP||⟨1 -1 1⟩BCC
Between alpha and beta
phases in Mg-Al aloys.
*Slide courtesy of S Mandal
65
Complex semi-coherent interfaces
•  It can often happen that an orientation relationship exists
despite the lack of an exact match.
•  Such is the case for the relationship between bcc and fcc
iron (ferrite and austenite).
Note limited atomic
match for the NW
relationship
*Slide courtesy of AD Rollett
66
Creating Coordinate Transformation Matrix for OR
Let’s say you are given an orientation relationship and you want to know the
rotation and change in volume between the two crystals. The coordinate
transformation matrix, J, contains both pieces of information.
Given:
(hkl)α//(h’k’l’)β, [uvw]α//[u’v’w’]β
First, define which phase is the parent phase and which is the derivative phase
(e.g. α→β)
m = (h k l )α (h' k ' l ')β
Define:
k = [u v w]α
g = [r
s
[u '
t ]α [r '
v' w']β
s ' t ']β
Where did [rst]α and [r’s’t’]β come from?
Can either be specified in OR or found by cross product of normal vector, n, for
(hkl) and direction vector, b, for [uvw]. (See earlier lectures for defining orientation
matrix from (hkl)[uvw].)
**k, m, and g relate the length of the lattice vectors of the two crystals.
67
Creating Coordinate Transformation Matrix for OR
Recall from previous lectors, each column represents the components of a basis
vector of α in β, in [uvw], [rst], (hkl) order.
To find J for α→β:
β
⎛ u' ⋅ k r' ⋅ g h' ⋅ m ⎞
⎛u
⎜
⎟
⎜
⎜ v' ⋅ k s' ⋅ g k' ⋅ m ⎟ = J × ⎜ v
⎜ w' ⋅ k t' ⋅ g l' ⋅ m ⎟
⎜w
⎝
⎠
⎝
⎛ u' ⋅ k
⎜
J = ⎜ v' ⋅ k
⎜ w' ⋅ k
⎝
α
r h⎞
⎟
s k⎟
t l ⎟⎠
r' ⋅ g h' ⋅ m ⎞ ⎛ u r h ⎞
⎟ ⎜
⎟
s' ⋅ g k' ⋅ m ⎟ × ⎜ v s k ⎟
t' ⋅ g l' ⋅ m ⎟⎠ ⎜⎝ w t l ⎟⎠
−1
Coordinate Transformation Matrix for OR to Strain
and Rotation Matrices
68
Within the transformation matrix are stretch, P, and rotation, U,
matrices. When multiplied together these matrices recover the
transformation matrix. J=UP
To find the stretch and rotation matrices use a polar decomposition†.
P = J *J ,
and
J * denotes the conjugate transpose of J
U = J ⋅ P −1
Per usual, transform rotation matrix, U, to other notations for other
uses
**Use MATLAB (function “poldec”) or some other computation
package to make this calculation easier**
†See,e.g.,AnIntroduc8ontoCon8nuumMechanics,M.Gur8n
Coordinate Transformation Matrix for OR to Strain
and Rotation Matrices (Example)
69
Given the Kurdjumov-Sachs OR for ferrite, α, and austenite, γ, below,
find the rotation matrix.
[1
[1
1 1]γ || [0 1 1]α
k=
aγ 3
] [
g=
aα 2
⎛0
⎜
⎜k
⎜⎜
⎝k
g
g
g
]
0 1 γ || 1 1 1 α
aγ 2
aα 3
[1
m=
] [
]
2 1 γ || 2 1 1 α
aγ 6
⎛1 1 1 ⎞
2m ⎞
⎟
⎜
⎟
m ⎟ = J × ⎜1 0 2 ⎟
⎟
⎜⎜
⎟⎟
m ⎟⎠
1
1
1
⎝
⎠
⎛ 3 g + 2m
4m
3g + 2m ⎞
⎜
⎟
1
J = ⎜ 2k + 3 g − m 2k + 2m 2k − 3 g − m ⎟
6⎜
⎜ 2k − 3 g + m 2k − 2m 2k + 3 g + m ⎟⎟
⎝
⎠
aα 6
=
aγ
aα
Coordinate Transformation Matrix for OR to Strain
and Rotation Matrices (Example)
Given:
aα = 0.28662 nm
aγ = 0.36551 nm
⎛ 0.9457 − 0.8502 − 0.0955 ⎞
⎜
⎟
J = ⎜ 0.8287 0.9457 − 0.2125 ⎟
⎜ 0.2125 0.0955
1.2538 ⎟⎠
⎝
Polar Decomposition
Rotation
⎛ 0.7416 − 0.6667 − 0.0749 ⎞
⎜
⎟
U = ⎜ 0.6598 0.7416 − 0.1667 ⎟
⎜ 0.1667 0.0749
0.9832 ⎟⎠
⎝
Stretch
⎛ 1.2752 0.0000 0.0000 ⎞
⎜
⎟
P = ⎜ 0.0000 1.2752 0.0000 ⎟
⎜ 0.0000 0.0000 1.2752 ⎟
⎝
⎠
Strain = P – 1
70
Summary
•  Specific OR exists between product and
parent phases for most of the
transformations
•  Well defined ORs also occur for thin film
deposition
•  Construction of misorientation matrix given
an OR is straight-forward given parallel (hkl)
[uvw]
•  Some variants of an OR may be preferred
over others depending on mechanism
71